1 Thank you for inviting us

• Kyle Baron
• Ahmed Elmokadem
• Mike Heathman

2 What can you do with mrgsolve?

3 About mrgsolve

• R package for simulation from ODE-based models
  • Free, OpenSource, GitHub, CRAN
• Language
  • Models written in C++ inside model specification format
  • General purpose solver: ODEPACK / DLSODA (FORTRAN)
    • Automatically detect and switch between non-stiff (Adams) and stiff (BDF) methods for solving the differential equations
  • Simulation workflow in R
• Hierarchical (population) simulation
  • ID, \(\eta\), \(\varepsilon\)
• Integrated PK functionaility
  • Bolus, infusion, F, ALAG, SS etc, handled under the hood
  • 1- and 2-cmt PK models in closed-form
• Extensible using R, C++, Rcpp, boost, RcppArmadillo
• R is it’s natural habitat

4 mrgsolve started as QSP modeling tool

• Motivation: large bone/mineral homeostatsis model (CaBone)
• History using
  • Berkeley Madonna
  • WinBUGS
  • NONMEM (attempted)
• 2010: write R front end to deSolve
• 2012: write C++ interface to DLSODA
• Develop dosing / event capability
• More recently, expose functionality provided by
  • Rcpp - vectors, matrices, functions, environments, random numbers
  • boost - numerical tools in C++
  • users’ own C++ code (functions, data structures, classes)
• Translator from SBML to mrgsolve using R bindings to libSBML

5 Orientation

• https://CRAN.R-project.org/package=mrgsolve

• GitHub site: https://github.com/metrumresearchgroup/mrgsolve

• Discussion: https://github.com/metrumresearchgroup/mrgsolve/discussions

• Issues: https://github.com/metrumresearchgroup/mrgsolve/issues

• mrgsolve website: https://mrgsolve.org

• User Guide: https://mrgsolve.org/user_guide

• Blog: https://mrgsolve.org/blog

• Vignettes: https://mrgsolve.org/vignettes

6 R for Data Science

https://r4ds.had.co.nz/

7 What we will cover today

1. Three basic workflows
2. Loading the model into R
3. Event objects
4. Data sets
5. Model specification - code together
6. Whatever you ask about

Emphasis is on getting you running your own simulations today.

8 A basic simulation with mrgsolve

mod %>% ev(amt = 100, ii = 24, addl = 3) %>% mrgsim() %>% plot()

9 A basic simulation with mrgsolve

mod %>% ev(amt = 100, ii = 24, addl = 3) %>% mrgsim() %>% plot()

• mod: the model object
  • Ok … where did that come from?
• ev(amt = 100, …) : the intervention
  • An event in this example
• mrgsim(): actually do the simulation
• plot(): do something with the simulation
  • plot, mutate, as_tibble etc …

• model %>% intervention %>% Go! %>% take-a-look

10 What’s coming …

• model %>% intervention %>% options %>% Go! %>% ...

• model %>% intervention %>% population %>% Go! %>% ...

• model %>% data-set %>% Go! %>% ...

• where data-set = intervention + population

• For now, let’s get this part down

• model %>% intervention %>% Go! %>% take-a-look

11 Why do we use %>% ?

What happens first in this operation?

mean(sqrt(seq(4)))

12 Pipelines

mean(sqrt(seq(4)))

4  %>% seq() %>% sqrt() %>% mean()

Better.

4  %>% 
  seq(.) %>% 
  sqrt(.) %>% 
  mean(., na.rm = TRUE)

mod %>% some_intervention() %>% simulate() %>% post_process()

13 The model object

model %>% ...

• I (almost) always call the model object mod in the documention / examples

• All the information about the model we need to know to run the simulation

• Distinct from the intervention, the population, the summary

• But the model does know about output time, random effects

14 Take a look: overview

mod

-----------------  source: pk1.cpp  -----------------

  project: /Users/kyleb/ghe...gsolve/models
  shared object: pk1-so-10d6a5b45d9a0 

  time:          start: 0 end: 192 delta: 0.2
                 add: <none>

  compartments:  EV CENT [2]
  parameters:    CL V KA [3]
  captures:      CP [1]
  omega:         0x0 
  sigma:         0x0 

  solver:        atol: 1e-08 rtol: 1e-08 maxsteps: 20k
------------------------------------------------------

15 Take a look: parameters (really important)

param(mod)

 Model parameters (N=3):
 name value . name value
 CL   1     | V    20   
 KA   1     | .    .    

• Parameters get a name
• Names and number of parameters gets fixed at compile time
• All parameters have a value
• Value can be modified after compile time

16 Take a look: compartments

init(mod)

 Model initial conditions (N=2):
 name       value . name     value
 CENT (2)   0     | EV (1)   0    

• Every compartment gets a name
• Every compartment gets an initial condition

17 Where did mod come from?

mod <- mread("simple", "model")

mod <- mread("<model-name>", "<project-directory>")

• By default mrgsolve looks for the code in the file
  • model-name.cpp in
  • project-directory
• mread demo

19 Model build directory (soloc)

It can be helpful to set

options(mrgsolve.soloc = "my_build_directory")

Equivalent to

mod <- mread("simple", here("day1/model"), soloc = "my_build_directory")

20 Internal model library

mod <- mread("<first-argument>", "<second-argument>")

21 Internal model library

mod <- mread("<first-argument>", "<second-argument>")

mod <- mread("<first-argument>", modlib())

modlib()

. [1] "/Users/kyleb/ghe/courses/pbpk-at-lilly-IU/rlibs/mrgsolve/models"

22 Internal model library

mod <- mread("effect", modlib())

mod

. 
. 
. ----------------  source: effect.cpp  ----------------
. 
.   project: /Users/kyleb/ghe...gsolve/models
.   shared object: effect-so-21f170c05f64 
. 
.   time:          start: 0 end: 36 delta: 0.1
.                  add: <none>
. 
.   compartments:  GUT CENT PERIPH CE [4]
.   parameters:    VC KA K10 K12 K21 E0 EMAX EC50 KEO [9]
.   captures:      EFFECT CP [2]
.   omega:         0x0 
.   sigma:         0x0 
. 
.   solver:        atol: 1e-08 rtol: 1e-08 maxsteps: 20k
. ------------------------------------------------------

23 Try this now

Open R script in Rstudio then …

library(mrgsolve)

mod <- mread("pbpk", modlib())

mod <- modlib("pbpk")

24 Your turn

• File name: day1/exercises/warm_up.R

• Section name: Warm Up

25 Event objects

• model %>% intervention %>% Go! %>% take-a-look

• Event object = quick / easy way to implement dose or other intervention

e <- ev(amt = 100) 

e

. Events:
.   time amt cmt evid
. 1    0 100   1    1

• Defaults: time, evid, cmt

26 Three ways to invoke

Inline

mod %>% ev(amt = 100) %>% mrgsim()

Object via pipeline

e <- ev(amt = 100)

mod %>% ev(e) %>% mrgsim()

As argument

mod %>% mrgsim(events = e)

27 What to include in ev(...)

• time event time
• cmt Event compartment
• amt Dose amount
• ii Inter-dose interval
• addl Additional doses to administer
• rate Infusion rate
• ss Set to 1 to advance to steady state
• evid Event id
• ID Subject ID (use multiple ids - ID=1:10)

28 Interventions and corresponding evid

• Bolus dosing (evid 1, with rate==0)
• Zero order infusion (evid 1, with rate > 0)
• Other type event (evid 2)
  • This also forces solver reset
• Compartment reset (evid 3)
• Reset and dose (evid 4)
• Replace the amount in a specific compartment (evid 8)

29 Your Turn

• File name: day1/exercises/pkpk-csa.R

30 Create complex events - 1

What’s going to happen?

e1 <- ev(amt = 200) 

e2 <- ev(amt = 100, time = 24, ii = 24, addl = 4)

c(e1, e2)

31 Create complex events - 1

What’s going to happen?

32 Create complex events - 1

Combine two events

e1 <- ev(amt = 200) 

e2 <- ev(amt = 100, time = 24, ii = 24, addl = 4)

c(e1, e2)

. Events:
.   time amt cmt evid ii addl
. 1    0 200   1    1  0    0
. 2   24 100   1    1 24    4

33 Create complex events - 2

What’s going to happen?

e1 <- ev(amt = 200, ii = 12, addl = 2) 

e2 <- ev(amt = 100, ii = 24, addl = 4)

seq(e1, e2)

34 Create complex events - 2

What’s going to happen?

35 Create complex events - 2

Put two events in a sequence

e1 <- ev(amt = 200, ii = 12, addl = 2) 

e2 <- ev(amt = 100, ii = 24, addl = 4)

seq(e1, e2)

. Events:
.   time amt ii addl cmt evid
. 1    0 200 12    2   1    1
. 2   36 100 24    4   1    1

36 Create complex events - 3

What is going to happen?

e1 <- ev(amt = 200) 

e2 <- ev(amt = 100, ii = 24, addl = 4)

seq(e1, wait = 36, e2)

37 Create complex events - 3

What is going to happen?

38 Create complex events - 3

Wait before starting the next part of the regimen

e1 <- ev(amt = 200) 

e2 <- ev(amt = 100, ii = 24, addl = 4)

seq(e1, wait = 36, e2)

. Events:
.   time amt ii addl cmt evid
. 1    0 200  0    0   1    1
. 2   36 100 24    4   1    1

39 Your Turn

• File Name: day1/examples/pbpk-rifampin-ddi.R

40 Event objects are just data frames

as.data.frame(e1)

.   time amt cmt evid
. 1    0 200   1    1

• We will use a data_set later on for populations

• Event objects are convenient

  • Constructor
  • Operations

41 Simulate

model %>% intervention %>% Go!

e <- ev(amt = 100)

mod %>% ev(e) %>% mrgsim()

. Model:  effect 
. Dim:    362 x 8 
. Time:   0 to 36 
. ID:     1 
.     ID time    GUT   CENT PERIPH     CE EFFECT     CP
. 1:   1  0.0   0.00  0.000 0.0000 0.0000  157.0  0.000
. 2:   1  0.0 100.00  0.000 0.0000 0.0000  157.0  0.000
. 3:   1  0.1  91.21  8.443 0.1552 0.2225  155.7  3.460
. 4:   1  0.2  83.19 15.502 0.5817 0.8048  152.8  6.353
. 5:   1  0.3  75.88 21.354 1.2272 1.6382  149.6  8.752
. 6:   1  0.4  69.21 26.156 2.0463 2.6356  146.6 10.719
. 7:   1  0.5  63.13 30.046 3.0001 3.7278  144.1 12.314
. 8:   1  0.6  57.58 33.146 4.0553 4.8607  142.2 13.584

mrgsim(mod, events = e)

42 What would you like to “fix” in this plot?

mod %>% ev(amt = 100) %>% mrgsim() %>% plot(CP~time)

43 Simulation end time

mod %>% ev(amt = 100) %>% mrgsim(end = 48) %>% plot(CP~time)

44 Simulation time step

mod %>% ev(amt = 100) %>% mrgsim(end = 48, delta = 0.1) %>% plot(CP~time)

45 Simulation time grid

• To form a sequence of times
  • start usually 0
  • end time of last observation
  • delta output time step
• Additional times to simulate
  • add ad-hoc numeric vector

stime(mod)

. [1]  0  6 12 18

46 We’re stil working on this setup

model %>% intervention %>% Go! %>% take-a-look

model:

• Load a model with mread or mread_cache
• Use the internal library with mread("<model-name>", modlib())
• Check model parameters with param(mod)
• Check model initial conditions with init(mod)
• View model code with see(mod)

intervention:

• ev(...): amt, cmt, time, ii, addl, rate
• Different ways to combine event objects

47 Working with simulated output

model %>% intervention %>% Go! %>% take-a-look

48 Plot

mod <- mread_cache("effect", modlib())

mod %>% ev(amt = 100) %>% mrgsim() %>% plot()

49 Plot

mod %>% ev(amt = 100) %>% mrgsim() %>% 
  plot(CP + EFFECT~., col = "firebrick")

50 Pipeline to dplyr functions

mod %>% mrgsim() %>% mutate(arm = 1)

. # A tibble: 361 × 9
.       ID  time   GUT  CENT PERIPH    CE EFFECT    CP   arm
.    <dbl> <dbl> <dbl> <dbl>  <dbl> <dbl>  <dbl> <dbl> <dbl>
.  1     1   0       0     0      0     0    157     0     1
.  2     1   0.1     0     0      0     0    157     0     1
.  3     1   0.2     0     0      0     0    157     0     1
.  4     1   0.3     0     0      0     0    157     0     1
.  5     1   0.4     0     0      0     0    157     0     1
.  6     1   0.5     0     0      0     0    157     0     1
.  7     1   0.6     0     0      0     0    157     0     1
.  8     1   0.7     0     0      0     0    157     0     1
.  9     1   0.8     0     0      0     0    157     0     1
. 10     1   0.9     0     0      0     0    157     0     1
. # … with 351 more rows

51 Controlling output - request

mod %>% mrgsim() %>% head(n = 2)

.   ID time GUT CENT PERIPH CE EFFECT CP
. 1  1  0.0   0    0      0  0    157  0
. 2  1  0.1   0    0      0  0    157  0

mod %>% Req(CP) %>% mrgsim() %>% head(n = 2)

.   ID time CP
. 1  1  0.0  0
. 2  1  0.1  0

52 Controlling output - obsonly

mod %>% ev(amt=1) %>% mrgsim() %>% head(n = 2)

.   ID time GUT CENT PERIPH CE EFFECT CP
. 1  1    0   0    0      0  0    157  0
. 2  1    0   1    0      0  0    157  0

mod %>% ev(amt=1) %>% obsonly() %>% mrgsim() %>% head(n = 2)

.   ID time       GUT       CENT      PERIPH          CE   EFFECT         CP
. 1  1  0.0 0.0000000 0.00000000 0.000000000 0.000000000 157.0000 0.00000000
. 2  1  0.1 0.9121052 0.08443153 0.001551552 0.002224565 156.9866 0.03460309

53 Updating the model object

54 Update

On the fly

mod %>% update(end = 72) %>% mrgsim()

Persistent

mod <- update(mod, end = 72)

mrgsim will call update for you (on the fly)

mod %>% mrgsim(end = 72)

55 What else can I update?

• Time
  • start, end, delta, add
• Parameters and compartment initial values
• Solver settings
  • atol, rtol
  • hmax, maxsteps, mxhnil, ixpr
• $OMEGA, $SIGMA
• tscale (rescale the output time)
• digits

mod %>% update(rtol = 1E-12) %>% ...

mod %>% mrgsim(rtol = 1E-12) %>% ...

56 Updating parameters

• Demo

mod <- param(mod, CL = 1.5)

57 Add a population element to the simulation

• model %>% intervention %>% Go! %>% take-a-look

• model %>% intervention %>% population %>% Go! %>% take-a-look

58 One population simulation with mrgsolve

mod %>%
ev(amt = 100, ii = 24, addl = 3) %>%
idata_set(pop) %>%
mrgsim() %>% plot()

59 idata_set takes in individual-level data

• ID - one per row
• Typically parameters in columns

head(pop, n = 3)

.   ID   CL   VC     KA KOUT IC50 FOO
. 1  1 1.05 47.8 0.8390 2.45 1.28   4
. 2  2 0.73 30.1 0.0684 2.51 1.84   6
. 3  3 2.82 23.8 0.1180 3.88 2.48   5

length(unique(pop$ID))

. [1] 10

• This tells mrgsolve to simulate 10 units or individuals

60 pop and mod are connected via parameters

head(pop, n = 3)

.   ID   CL   VC     KA KOUT IC50 FOO
. 1  1 1.05 47.8 0.8390 2.45 1.28   4
. 2  2 0.73 30.1 0.0684 2.51 1.84   6
. 3  3 2.82 23.8 0.1180 3.88 2.48   5

param(mod)

. 
.  Model parameters (N=3):
.  name value . name value
.  CL   1     | V    20   
.  KA   1     | .    .

61 What else can we do with idata?

Batches of simulations or sensitivity analyses

idata <- expand.idata(CL = seq(0.5, 1.5, 0.25))
idata

.   ID   CL
. 1  1 0.50
. 2  2 0.75
. 3  3 1.00
. 4  4 1.25
. 5  5 1.50

62 Note: this is the event + idata_set configuration

mod %>%
  idata_set(idata) %>%
  ev(amt = 100, ii = 24, addl = 2) %>%
  mrgsim(end = 120) %>% plot(log(CP) ~ .)

63 Your turn

• Description: sensitivity analysis in a PBPK model

• File name: exercises/sensi_pbpk.R

64 data_set is the dosing equivalent to idata_set

data <- expand.ev(amt = c(100, 300, 1000), ii = 24, addl = 3)

head(data)

.   ID time  amt ii addl cmt evid
. 1  1    0  100 24    3   1    1
. 2  2    0  300 24    3   1    1
. 3  3    0 1000 24    3   1    1

65 Note: this is data_set configuration

mod %>%
  data_set(data) %>%
  mrgsim(end = 120) %>% plot(log(CP) ~ .)

66 data_set can also carry parameters

data <- expand.ev(
  amt = c(100,300,1000),
  ii = 24, addl = 3,
  CL = seq(0.5,1.5, 0.5)
)

head(data)

.   ID time  amt ii addl cmt evid  CL
. 1  1    0  100 24    3   1    1 0.5
. 2  2    0  300 24    3   1    1 0.5
. 3  3    0 1000 24    3   1    1 0.5
. 4  4    0  100 24    3   1    1 1.0
. 5  5    0  300 24    3   1    1 1.0
. 6  6    0 1000 24    3   1    1 1.0

67 carry_out or carry.out

Copy from the input data to the output data

mod %>% carry_out(dose) %>% data_set(data, dose==300) %>% mrgsim()

. Model:  pk1 
. Dim:    2886 x 4 
. Time:   0 to 192 
. ID:     3 
.     ID time dose     CP
. 1:   2  0.0  300  0.000
. 2:   2  0.0  300  0.000
. 3:   2  0.2  300  2.712
. 4:   2  0.4  300  4.919
. 5:   2  0.6  300  6.712
. 6:   2  0.8  300  8.167
. 7:   2  1.0  300  9.345
. 8:   2  1.2  300 10.296

68 data_set and ev

head(data, n = 3)

.   ID time  amt ii addl cmt evid  CL dose
. 1  1    0  100 24    3   1    1 0.5  100
. 2  2    0  300 24    3   1    1 0.5  300
. 3  3    0 1000 24    3   1    1 0.5 1000

ev(amt = 100, ii = 24, addl = 3)

. Events:
.   time amt ii addl cmt evid
. 1    0 100 24    3   1    1

69 We have now seen 3 simulation setups

• mod + ev = ?
• mod + idata_set + ev = ?
• mod + data_set = ?

70 Wait a minute …

head(data)

.   ID time  amt ii addl cmt evid  CL dose
. 1  1    0  100 24    3   1    1 0.5  100
. 2  2    0  300 24    3   1    1 0.5  300
. 3  3    0 1000 24    3   1    1 0.5 1000
. 4  4    0  100 24    3   1    1 1.0  100
. 5  5    0  300 24    3   1    1 1.0  300
. 6  6    0 1000 24    3   1    1 1.0 1000

71 Generating data sets

Recall that data sets are just plain old data.frames in R. Feel free to code these up in any way that is convenient for you.

In our experience, the following helper functions cover many (not every) common needs for building the data sets.

• expand.ev
• ev_rep
• as_data_set
• ev_assign

72 expand.ev

Like expand.grid

expand.ev(ID = 1:2, amt = c(100,200))

.   ID time amt cmt evid
. 1  1    0 100   1    1
. 2  2    0 100   1    1
. 3  3    0 200   1    1
. 4  4    0 200   1    1

73 ev_rep

e <- ev(amt = 100, ii = 12, addl = 14)
ev_rep(e, ID = seq(5))

.     ID time amt ii addl cmt evid
. 1    1    0 100 12   14   1    1
. 1.1  2    0 100 12   14   1    1
. 1.2  3    0 100 12   14   1    1
. 1.3  4    0 100 12   14   1    1
. 1.4  5    0 100 12   14   1    1

e <- ev(amt = 100, ii = 12, addl = 14, ID = seq(5))

74 ev_rep

e <- seq(ev(amt = 100), wait = 36, ev(amt = 50, ii = 24, addl = 4))
e

. Events:
.   time amt ii addl cmt evid
. 1    0 100  0    0   1    1
. 2   36  50 24    4   1    1

ev_rep(e, ID = seq(2))

.     ID time amt ii addl cmt evid
. 1    1    0 100  0    0   1    1
. 2    1   36  50 24    4   1    1
. 1.1  2    0 100  0    0   1    1
. 2.1  2   36  50 24    4   1    1

75 as_data_set

data <- as_data_set(
  ev(amt = 100, ii = 12, addl = 19, ID = 1:2),
  ev(amt = 200, ii = 24, addl = 9,  ID = 1:3),
  ev(amt = 150, ii = 24, addl = 9,  ID = 1:4)
)

.   ID time cmt evid amt ii addl
. 1  1    0   1    1 100 12   19
. 2  2    0   1    1 100 12   19
. 3  3    0   1    1 200 24    9
. 4  4    0   1    1 200 24    9
. 5  5    0   1    1 200 24    9
. 6  6    0   1    1 150 24    9
. 7  7    0   1    1 150 24    9
. 8  8    0   1    1 150 24    9
. 9  9    0   1    1 150 24    9

76 ev_assign

pop <- expand.idata(GROUP = c(1,2), ID = 1:3)
head(pop)

.   ID GROUP
. 1  1     1
. 2  2     2
. 3  3     1
. 4  4     2
. 5  5     1
. 6  6     2

e1 <- ev(amt = 100, ii = 24, addl = 9)
e2 <- mutate(e1, amt = 200)

data <- ev_assign(
  l = list(e1,e2), 
  idata = pop, 
  evgroup = "GROUP",
  join = TRUE
)

head(data)

.   time amt ii addl cmt evid ID GROUP
. 1    0 100 24    9   1    1  1     1
. 2    0 200 24    9   1    1  2     2
. 3    0 100 24    9   1    1  3     1
. 4    0 200 24    9   1    1  4     2
. 5    0 100 24    9   1    1  5     1
. 6    0 200 24    9   1    1  6     2

77 Your turn

• Description: simulate GCSF PK/PD profiles for weight-based dosing

• File name: exercises/gcsf.R

78 Your turn

• Description: Population PK of azithromycin

• File name: exercises/population_azithromycin.R

79 Your turn

• Description: simulate PK profiles using empirical Bayes estimates from a meropenem model fit in NONMEM

• File name: exercises/sim_ebe.R

80 Model specification

• Assign name=value pairs
  • like CL = 1.25
  • “parameters”
• Define compartment names and numbers
• Derived inputs that influence ODE system
  • like ke = CL/V
• Write ODEs
• Derived outputs
  • Signal that we want this output

81 Let’s make this model

• What are the parameters?
• What are the compartments?
• What are the differential equations?

82 Code up the model

83 Block $PARAM [R] declare and initialize model parameters

$PARAM CL = 1, VC = 20, KA=1.2

• Model “parameters” with name = value format
• Separate by , or <newline>
• Use any name except for words in mrgsolve:::reserved()
• Values are evaluated by R parser
• Multiple blocks allowed
• Use name anywhere in the model
• Updatable in R but read-only in the model specification file
  • Consider $FIXED if appropriate
• Most often, you will want to match names in $PARAM with the names in your input data sets

84 Block $CMT [txt] declare model compartments

• Specify name and number of compartments in the model
• Use any name except for those listed in mrgsolve:::reserved()
• See also or $INIT

Example $CMT

$CMT GUT CENT

Example $INIT

$INIT GUT = 0, CENT = 10

85 Block $ODE [C++] write differential equations

• Every compartment needs an equation
• Form:dxdt_CMT = ratein - rateout;
• Use CMT for compartment amounts, parameters, or other variables
• This block of code gets called repeatedly so be wise

$ODE
dxdt_GUT  = -KA*GUT;
dxdt_CENT =  KA*GUT - (CL/VC)*CENT;

86 Derive new variables in your model

• Your model code is written in C++, which is a language
• When you want a new variable, you must say what it is
• $MAIN, $ODE, $TABLE, $GLOBAL

To get a numeric variable, use double

double x = 5.4;

Other types you might use

bool cured = false;
int i = 2;

If in doubt, use double; it’s what you want most of the time.

87 Block $MAIN [C++] covariate model & initials

For example

$MAIN
double CL = TVCL*pow(WT/70,0.75);
double VC = TVVC*pow(0.86,SEX);
RESP_0 = KIN/KOUT;

In this example

• TVCL, KIN, KOUT, WT, SEX were declared in $PARAM
• There is a response compartment RESP

88 C++ expressions and functions

if(a == 2) b = 2;
if(b <= 2) {
  c=3;
} else {
  c=4;
}
double d = pow(base,exponent);
double e = exp(3);
double f = fabs(-4);
double g = sqrt(5);
double h = log(6);
double i = log10(7);
double j = floor(4.2);
double k = ceil(4.2);

Lots of help on the web http://en.cppreference.com/w/cpp/numeric/math/tgamma

89 Preprocessor directives (#define)

$GLOBAL
#define CP (CENT/VC)
#define g 9.8
$ODE
double INH = CP/(EC50+CP);
dxdt_RESP = KIN*(1-INH) - KOUT*RESP;
$CAPTURE CP

90 Introduction to POPULATION simulation

• Like NONMEM, use $OMEGA and $SIGMA to define covariance matrices for IIV and RUV, respectively

• Like NONMEM, use ETA(n) and EPS(m) for realized random effects drawn from $OMEGA and $SIGMA respectively

  • BUT …. use labeled ETAs instead

• Like NONMEM, mrgsolve recognizes ID as a subject indicator for simulating a new ETA

• Like NONMEM, mrgsolve allows you to write an error model as a function of EPS(m) and any other calculated value in your model

91 Block $OMEGA and $SIGMA [txt]

• Specify random effect variance/covariance matrices
  • \(\eta_i \sim N\left(0, \Omega \right)\)
  • \(\varepsilon_j \sim N \left(0, \Sigma \right)\)
• NONMEM-style input - lower triangle
• Multiple $OMEGA and $SIGMA are allowed and each may be named
• How you enter these blocks defines a that mrgsolve will enforce later on
• Diagonal matrix (3x3 in this example)

$OMEGA 0.1 0.2 0.3

92 More $OMEGA and $SIGMA

• Options are @ delimited & must be on different line from the matrix data
• Block matrix (2x2 in this example)

$OMEGA @block
0.1 0.0947 0.2

• Block matrix with correlations (assumes @block)

$OMEGA @cor
0.1 0.67 0.2

93 Block $OMEGA and $SIGMA

• Multiple $OMEGA and $SIGMA are allowed; each may be named

$OMEGA @name PK
0 0 0
$OMEGA @name PD
0 0

Users are encouraged to add labels

$OMEGA @name PK @labels ECL EVC EKA
0 0 0

94 About @ macros

• Used to indicate options associated with different code blocks
• Most options are for $OMEGA and $SIGMA
• @ should appear at the start of the line and the entire line is reserved for options
• A line may contain multiple options: @cor @name PK

Two forms:

• Boolean: @block means block=TRUE
• Name/value: @labels ECL EVC means labels=c("ECL", "EVC")

95 Closed-form PK models with $PKMODEL

• Set ncmt to 1 or 2
• Set depot to TRUE for extravascular dosing compartment
• Set trans to change the names of required parameters
• Use $CMT to declare 1 to 3 compartments as appropriate

$PKMODEL cmt = "GUT CENT PERIPH", depot=TRUE
$PARAM CL = 1 , V2 = 30, Q = 8, V3 = 400, KA=1.2

96 More $PKMODEL

$PKMODEL ncmt=1, depot=TRUE
$CMT GUT CENT
$PARAM TVCL = 1 , TVV = 30, KA=1.2
$OMEGA @labels ECL EVC
1 1
$MAIN
double CL = TVCL*exp(ECL);
double V  = TVV *exp(EVC);

97 Bioavailability, dosing lag time, and infusions

• To change the bioavability of doses administered to a compartment, set F_CMT in $MAIN

• To add a lagtime for doses administered to a compartment, set ALAG_CMT in $MAIN

• To set the duration of infusion, set D_CMT in $MAIN

  • Also, use rate = -2 in your data set or event object

• To set the infusion rate, set R_CMT in $MAIN

  • ALso, use rate = -1 in your data set or event object

98 Functions

• Two arguments: x, and y
• Pass arguments by name
• Return one thing z

add <- function(x,y) {
  z <- x + y
  return(z)
}

add(x = 2, y = 3)

. [1] 5

99 Functions

• Arguments can be matched by position
• Calling return() is not strictly necessary

add <- function(x, y) {
  x + y
}

add(2,3)

. [1] 5

100 Functions

• Arguments can have default values

add <- function(x, y=3) {
  x + y
}

add(2)

. [1] 5

101 Sensitivity analysis

• Ad-hoc
  • Sequential: systematically vary the parameter value
  • Random: sample from a parametric or non-parametric distribution
• Local
  • small changes around a nominal value; one at a time
• Global
  • explore the whole parameter space; includes interactions

102 Local sensitivity analysis

We use the lsa() function from the mrgsim.sa package

library(mrgsim.sa)

?lsa

sens <- lsa(model_object, parameter_names, output_variables)

103 First, look at the result

104 A simulation scenario of interest

mod <- modlib("irm1", end = 72, delta = 0.1)

mod %>%
  ev(amt = 100) %>% 
  mrgsim(end = 72, delta = 0.1) %>% 
  plot()

105 lsa

sens <- lsa(
  mod = ev(mod, amt = 100),
  par = "CL,V2,Q,KA", 
  var = "CP"
)

head(sens, n=3)

. # A tibble: 3 × 5
.    time dv_name dv_value p_name     sens
.   <dbl> <chr>      <dbl> <chr>     <dbl>
. 1   0   CP         0     CL      0      
. 2   0   CP         0     CL      0      
. 3   0.1 CP         0.472 CL     -0.00254

106 The default plot method

plot(sens)

107 Global senstivity analysis

• All the parameters vary at once
• Investigate the whole parameter space
  • Or reasonable parameter space
• Evaluate both individual parameters as well as interactions between parameters
• Much greater computational burden

We’ll look at Sobol’s method

108 Global senstivity analysis

• Sobol Sensitivity Analysis: A Tool to Guide the Development and Evaluation of Systems Pharmacology Models.
  • Zhang XY, Trame MN, Lesko LJ, Schmidt S.
  • CPT Pharmacometrics Syst Pharmacol. 2015 Feb;4(2):69-79. doi: 10.1002/psp4.6.
  • PMID: 27548289
  • https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5006244/

109 Global senstivity analysis

• Sobol Sensitivity Analysis: A Tool to Guide the Development and Evaluation of Systems Pharmacology PBPK Models.
  • Zhang XY, Trame MN, Lesko LJ, Schmidt S.
  • CPT Pharmacometrics Syst Pharmacol. 2015 Feb;4(2):69-79. doi: 10.1002/psp4.6.
  • PMID: 27548289
  • https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5006244/

110 Sobol

• Variabillity in the output is decomposed, relating to variability in different inputs or combinations of inputs
• We’ll generate random samples of model parameters of interest and pass them off to sensitivity::sobol2007
• From our experience, put some reaonable limit parameter space
• Also, create a function that accepts a set of parameters and returns a simulated statistic of interest (like AUC)